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Properties

Label 2-1216-19.7-c1-0-13
Degree $2$
Conductor $1216$
Sign $0.0977 - 0.995i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (1.5 + 2.59i)5-s + (1 − 1.73i)9-s + 4·11-s + (−2.5 + 4.33i)13-s + (−1.5 + 2.59i)15-s + (2.5 + 4.33i)17-s + (−4 − 1.73i)19-s + (0.5 − 0.866i)23-s + (−2 + 3.46i)25-s + 5·27-s + (1.5 − 2.59i)29-s + 4·31-s + (2 + 3.46i)33-s − 2·37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.670 + 1.16i)5-s + (0.333 − 0.577i)9-s + 1.20·11-s + (−0.693 + 1.20i)13-s + (−0.387 + 0.670i)15-s + (0.606 + 1.05i)17-s + (−0.917 − 0.397i)19-s + (0.104 − 0.180i)23-s + (−0.400 + 0.692i)25-s + 0.962·27-s + (0.278 − 0.482i)29-s + 0.718·31-s + (0.348 + 0.603i)33-s − 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.0977 - 0.995i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.163550452\)
\(L(\frac12)\) \(\approx\) \(2.163550452\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (4 + 1.73i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.5 + 4.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.5 - 11.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.5 - 7.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.09218622119851267971677131830, −9.185990198395943750064219674179, −8.577284145241727260502725660451, −7.16037949655193162309034997164, −6.59867913544136811142849482496, −6.03593189324504591406776830332, −4.49860334420986502597813157110, −3.85058148625644840414696704793, −2.76958147167755201392029308328, −1.65343395986600288155362959938, 0.973127705026115125339362395937, 1.92866026723187867845631256551, 3.16848134761799190016259149630, 4.62288120007408061392660709754, 5.16188938104293955838932428878, 6.19763187865265040010956589756, 7.13509691775498423330180611495, 8.047099642116034488546395432749, 8.618370361336913708281870799250, 9.623336740363681090136830860476

Graph of the $Z$-function along the critical line